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Concept# Vector (mathematics and physics)

Summary

In mathematics and physics, vector is a term that refers colloquially to some quantities that cannot be expressed by a single number (a scalar), or to elements of some vector spaces.
Historically, vectors were introduced in geometry and physics (typically in mechanics) for quantities that have both a magnitude and a direction, such as displacements, forces and velocity. Such quantities are represented by geometric vectors in the same way as distances, masses and time are represented by real numbers.
The term vector is also used, in some contexts, for tuples, which are finite sequences of numbers of a fixed length.
Both geometric vectors and tuples can be added and scaled, and these vector operations led to the concept of a vector space, which is a set equipped with a vector addition and a scalar multiplication that satisfy some axioms generalizing the main properties of operations on the above sorts of vectors. A vector space formed by geometric vectors is called a Euclidean vecto

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We discuss some properties of generative models for word embeddings. Namely, (Arora et al., 2016) proposed a latent discourse model implying the concentration of the partition function of the word vectors. This concentration phenomenon led to an asymptotic linear relation between the pointwise mutual information (PMI) of pairs of words and the scalar product of their vectors. Here, we first revisit this concentration phenomenon and prove it under slightly weaker assumptions, for a set of random vectors symmetrically distributed around the origin. Second, we empirically evaluate the relation between PMI and scalar products of word vectors satisfying the concentration property. Our empirical results indicate that, in practice, this relation does not hold with arbitrarily small error. This observation is further supported by two theoretical results: (i) the error cannot be exactly zero because the corresponding shifted PMI matrix cannot be positive semidefinite; (ii) under mild assumptions, there exist pairs of words for which the error cannot be close to zero. We deduce that either natural language does not follow the assumptions of the considered generative model, or the current word vector generation methods do not allow the construction of the hypothesized word embeddings.

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